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Project-Euler
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**9**

Problem 9 asks:

There exists exactly one Pythagorean triplet for which \(a + b + c = 1000\). Find the product \(abc\).

I solved this with pen and paper.

I already know a Pythagorean triplet \((a, b, c)\) can be written as \(k(2mn, m^2-n^2, m^2+n^2)\) where \(k, m, n \in \mathbb{N} \cdot m > n\) by Euclid’s formula, so using that fact:$$a + b + c = 1000$$ can be writen as, $$\begin{align}&2mn + m^2 - n^2 + m^2 + n^2 = 1000\\ \implies &m(m + n) = 500\\ \implies &m = 20, n = 5\end{align}$$ therefore, $$\begin{align}a &= 2\times20\times5 = 200\\ b &= 20^2 - 5^2 = 375\\ c &= 20^2 + 5^2 = 425\\ abc &= 200\times375\times425 = 31875000\end{align}$$